# PiecewisePareto_Match_Layer_Losses: Match a Tower of Expected Layers Losses In Pareto: The Pareto, Piecewise Pareto and Generalized Pareto Distribution

 PiecewisePareto_Match_Layer_Losses R Documentation

## Match a Tower of Expected Layers Losses

### Description

Matches the expected losses of a tower of reinsurance layers using a piecewise Pareto severity

### Usage

``````PiecewisePareto_Match_Layer_Losses(
Attachment_Points,
Expected_Layer_Losses,
Unlimited_Layers = FALSE,
Frequencies = NULL,
FQ_at_lowest_AttPt = NULL,
FQ_at_highest_AttPt = NULL,
TotalLoss_Frequencies = NULL,
minimize_ratios = TRUE,
Use_unlimited_Layer_for_FQ = TRUE,
truncation = NULL,
truncation_type = "lp",
dispersion = 1,
tolerance = 1e-10,
alpha_max = 100,
merge_tolerance = 1e-06,
RoL_tolerance = 1e-06
)
``````

### Arguments

 `Attachment_Points` Numeric vector. Vector containing the attachment points of consecutive layers in increasing order `Expected_Layer_Losses` Numeric vector. Vector containing the expected losses of layers xs the attachment points. `Unlimited_Layers` Logical. If `TRUE`, then `Expected_Layer_Losses[i]` contains the expected loss of `Inf` xs `Attachment_Points[i]`. If `FALSE` then `Expected_Layer_Losses[i]` contains the expected loss of the layer `Attachment_Points[i+1]` xs `Attachment_Points[i]` `Frequencies` Numeric vector. Expected frequencies excess the attachment points. The vector may contain NAs. If `NULL` then the function calculates frequencies. `FQ_at_lowest_AttPt` Numerical. Expected frequency excess `Attachment_Points[1]`. Overrules first entry in Frequencies. `FQ_at_highest_AttPt` Numerical. Expected frequency excess `Attachment_Points[k]`. Overrules last entry in Frequencies. `TotalLoss_Frequencies` Numeric vector. `TotalLoss_Frequencies[i]` is the frequency of total losses to layer `i` (i.e. `Attachment_Points[i+1] - Attachment_Points[i]` xs `Attachment_Points[i]`). `TotalLoss_Frequencies[i]` is the frequency for losses larger than or equal to `Attachment_Points[i+1]`, whereas `Frequencies[i]` is the frequency of losses larger than `Attachment_Points[i]`. `TotalLoss_Frequencies[i] > Frequencies[i+1]` means that there is a point mass of the severity at `Attachment_Points[i+1]`. `minimize_ratios` Logical. If `TRUE` then ratios between alphas are minimized. `Use_unlimited_Layer_for_FQ` Logical. Only relevant if no frequency is provided for the highest attachment point by the user. If `TRUE` then the frequency is calculated using the Pareto alpha between the last two layers. `truncation` Numeric. If `truncation` is not `NULL`, then the distribution is truncated at `truncation`. `truncation_type` Character. If `truncation_type = "wd"` then the whole distribution is truncated. If `truncation_type = "lp"` then a truncated Pareto is used for the last piece. `dispersion` Numerical. Dispersion of the claim count distribution in the resulting PPP_Model. `tolerance` Numeric. Numerical tolerance. `alpha_max` Numerical. Maximum alpha to be used for the matching. `merge_tolerance` Numerical. Consecutive Pareto pieces are merged if the alphas deviate by less than merge_tolerance. `RoL_tolerance` Numerical. Consecutive layers are merged if RoL decreases less than factor `1 - RoL_tolerance`.

### Value

A PPP_Model object that contains the information about a collective model with a Panjer distributed claim count and a Piecewise Pareto distributed severity. The object contains the following elements:

• `FQ` Numerical. Frequency in excess of the lowest threshold of the piecewise Pareto distribution

• `t` Numeric vector. Vector containing the thresholds for the piecewise Pareto distribution

• `alpha` Numeric vector. Vector containing the Pareto alphas of the piecewise Pareto distribution

• `truncation` Numerical. If `truncation` is not `NULL` and `truncation > max(t)`, then the distribution is truncated at `truncation`.

• `truncation_type` Character. If `truncation_type = "wd"` then the whole distribution is truncated. If `truncation_type = "lp"` then a truncated Pareto is used for the last piece.

• `dispersion` Numerical. Dispersion of the Panjer distribution (i.e. variance to mean ratio).

• `Status` Numerical indicator: 0 = success, 1 = some information has been ignored, 2 = no solution found

• `Comment` Character. Information on whether the fit was successful

### References

Riegel, U. (2018) Matching tower information with piecewise Pareto. European Actuarial Journal 8(2): 437–460

### Examples

``````AP <- Example1_AP
EL <- Example1_EL
PiecewisePareto_Match_Layer_Losses(AP, EL)
EL_unlimited <- rev(cumsum(rev(Example1_EL)))
PiecewisePareto_Match_Layer_Losses(AP, EL_unlimited, Unlimited_Layers = TRUE)
PiecewisePareto_Match_Layer_Losses(AP, EL, FQ_at_lowest_AttPt = 0.5)
Example1_FQ <- c(0.3, 0.15, 0.08, 0.02, 0.005)
PiecewisePareto_Match_Layer_Losses(AP, EL, Frequencies = Example1_FQ)

``````

Pareto documentation built on April 18, 2023, 9:10 a.m.